TY - JOUR A1 - Fackeldey, Konstantin A1 - Sikorski, Alexander A1 - Weber, Marcus T1 - Spectral Clustering for Non-Reversible Markov Chains JF - Computational and Applied Mathematics N2 - Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We also extend this spectral clustering method to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension non-reversible Markov chains can be analyzed. Furthermore, the chains do not need to have a positive stationary distribution. In addition to metastabilities, dominant cycles and sinks can also be identified. This novel method is called GenPCCA (i.e., generalized PCCA), since it includes the case of non-reversible processes. We also apply the method to real-world eye-tracking data. KW - Spectral clustering KW - Markov chain KW - Non-reversible KW - Schur decomposition KW - GenPCCA Y1 - 2018 U6 - https://doi.org/https://doi.org/10.1007/s40314-018-0697-0 VL - 37 IS - 5 SP - 6376 EP - 6391 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Fackeldey, Konstantin A1 - Weber, Marcus T1 - A Square Root Approximation of Transition Rates for a Markov State Model JF - SIAM. J. Matrix Anal. Appl. Y1 - 2013 U6 - https://doi.org/10.1137/120899959 VL - 34 IS - 2 SP - 738 EP - 756 ER - TY - JOUR A1 - Scharkoi, Olga A1 - Fackeldey, Konstantin A1 - Merkulow, Igor A1 - Andrae, Karsten A1 - Weber, Marcus A1 - Nehls, Irene T1 - Conformational Analysis of Alternariol on the Quantum Level JF - J. Mol. Model. Y1 - 2013 U6 - https://doi.org/10.1007/s00894-013-1803-2 VL - 19 IS - 6 SP - 2567 EP - 2572 ER - TY - GEN A1 - Bujotzek, Alexander A1 - Schütt, Ole A1 - Nielsen, Adam A1 - Fackeldey, Konstantin A1 - Weber, Marcus T1 - Efficient Conformational Analysis by Partition-of-Unity Coupling T2 - Math Chem N2 - Obtaining a sufficient sampling of conformational space is a common problem in molecular simulation. We present the implementation of an umbrella-like adaptive sampling approach based on function-based meshless discretization of conformational space that is compatible with state of the art molecular dynamics code and that integrates an eigenvector-based clustering approach for conformational analysis and the computation of inter-conformational transition rates. The approach is applied to three example systems, namely n-pentane, alanine dipeptide, and a small synthetic host-guest system, the latter two including explicitly modeled solvent. T3 - ZIB-Report - 13-58 KW - Markov State Models KW - Meshfree KW - Molecular Simulation KW - Partition of Unity Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42570 SN - 1438-0064 ER - TY - GEN A1 - Weber, Marcus A1 - Fackeldey, Konstantin A1 - Schütte, Christof T1 - Set-free Markov State Building N2 - Molecular dynamics (MD) simulations face challenging problems since the timescales of interest often are much longer than what is possible to simulate and even if sufficiently long simulation are possible the complex nature of the resulting simulation data makes interpretation difficult. Markov State Models (MSMs) help to overcome these problems by making experimentally relevant timescales accessible via coarse grained representations that also allows for convenient interpretation. However, standard set-based MSMs exhibit some caveats limiting their approximation quality and statistical significance. One of the main caveats results from the fact that typical MD trajectories repeatedly re-cross the boundary between the sets used to build the MSM which causes statistical bias in estimating the transition probabilities between these sets. In this article, we present a set-free approach to MSM building utilizing smooth overlapping ansatz functions instead of sets and an adaptive refinement approach. This kind of meshless discretization helps to overcome the recrossing problem and yields an adaptive refinement procedure that allows to improve the quality of the model while exploring state space and inserting new ansatz functions into the MSM. T3 - ZIB-Report - 17-10 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-62167 SN - 1438-0064 ER - TY - CHAP A1 - Durmaz, Vedat A1 - Fackeldey, Konstantin A1 - Weber, Marcus ED - Mode, Ch. T1 - A rapidly Mixing Monte Carlo Method for the Simulation of Slow Molecular Processes T2 - Applications of Monte Carlo Methods in Biology, Medicine and Other Fields of Science Y1 - 2011 PB - InTech ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf T1 - Quadrature and Implementation of the Weak Coupling Method T2 - MMM 2008: Proceedings of the 4th International Conference on Multiscale Materials Modelling, 27-31 October 2008, Tallahassee, Fla. Y1 - 2008 SP - 62 EP - 65 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf T1 - Weak Coupling Algorithms in Multiscale Simulations T2 - Proceedings of the third Inter. Confer. on Comp. Meth. for Coupled Problems in Science and Engrg Y1 - 2009 SP - 1023pp ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf ED - Griebel, Michael ED - Schweitzer, M. T1 - Numerical Validation of Constraints Based Multiscale Methods T2 - Lecture Notes in Computational Science and Engineering Y1 - 2010 SP - 141 EP - 154 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Krause, Dorian A1 - Krause, Rolf A1 - Lenzen, Christoph T1 - Coupling Molecular Dynamics and Continua with Weak Constraints JF - SIAM Multiscale Model. Simul. Y1 - 2011 U6 - https://doi.org//10.1137/100782097 N1 - also as MATHEON Preprint VL - 9 SP - 1459 EP - 1494 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf A1 - Schweitzer, M. ED - Griebel, Michael ED - Schweitzer, M. T1 - Stability of Energy Transfer in the Weak Coupling Method T2 - Meshfree Methods for Partial Differential Equations IV Y1 - 2008 VL - 65 SP - 111 EP - 121 ER - TY - JOUR A1 - Fackeldey, Konstantin T1 - Coupling Meshbased and Meshfree Methods by a Transfer Operator Approach JF - IJPAM Y1 - 2011 VL - 69 IS - 2 SP - 185 EP - 202 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - Multiscale Coupling in Function Space - Weak Coupling between Molecular Dynamics and Continuum Mechanics JF - International Journal for Numerical Methods in Engineering Y1 - 2009 U6 - https://doi.org/10.1002/nme.2626 VL - 79 IS - 12 SP - 1517 EP - 1535 ER - TY - THES A1 - Fackeldey, Konstantin T1 - The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics Y1 - 2009 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf ED - Widlund, Olof T1 - Solving Frictional Contact Problems with Multigrid Efficiency T2 - Proc.of the 16th International Conference on Domain Decomposition Methods Y1 - 2007 VL - 50 SP - 547 EP - 554 ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - Weak coupling in function space T2 - Proc. Appl. Math. Mech. Y1 - 2007 VL - 7 IS - 1 SP - 2020113pp ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Krause, Rolf T1 - CM/MD Coupling - A Function Space Oriented Multiscale-Coupling Approach T2 - Proc. Appl. Math. Mech. Y1 - 2008 VL - 8 IS - 1 SP - 10495pp ER - TY - CHAP A1 - Fackeldey, Konstantin A1 - Bujotzek, Alexander A1 - Weber, Marcus T1 - A meshless discretization method for Markov state models applied to explicit water peptide folding simulations T2 - Meshfree Methods for Partial Differential Equations VI Y1 - 2012 VL - 89 SP - 141 EP - 154 PB - Springer ER - TY - CHAP A1 - Fackeldey, Konstantin T1 - Multiscale Methods in Time and Space T2 - Progress in Industrial Mathematics at ECMI 2010 Y1 - 2012 VL - 17 SP - 619 EP - 626 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Klimm, Martina A1 - Weber, Marcus T1 - A Coarse Graining Method for the Dimension Reduction of the State Space of Biomolecules JF - Journal of Mathematical Chemistry Y1 - 2012 VL - 5 IS - 9 SP - 2623 EP - 2635 ER - TY - JOUR A1 - Krause, Dorian A1 - Fackeldey, Konstantin A1 - Krause, Rolf ED - Griebel, Michael T1 - A Parallel Multiscale Simulation Toolbox for Coupling Molecular Dynamics and Finite Elements JF - Singular Phenomena and Scaling in Mathematical Models Y1 - 2014 U6 - https://doi.org/10.1007/978-3-319-00786-1_14 SP - 327 EP - 346 PB - Springer International Publishing ER -